Related papers: Quantum Dimension and Quantum Projective Spaces

We develop a geometric approach to quantum mechanics based on the concept of the Tulczyjew triple. Our approach is genuinely infinite-dimensional and including a Lagrangian formalism in which self-adjoint (Schroedinger) operators are…

Various aspects of recent works on affine quantum group symmetry of integrable 2d quantum field theory are reviewed and further clarified. A geometrical meaning is given to the quantum double, and other properties of quantum groups.…

We study the quantum Riemannian geometry of quantum projective spaces of any dimension. In particular we compute the Riemann and Ricci tensors, using previously introduced quantum metrics and quantum Levi-Civita connections. We show that…

In the search of a mathematical basis for quantum mechanics, in order to render it self-consistent and rationally understandable, we find that the best approach is to adopt E. Cartan's way for discovering spinors; that is to start from…

Modified universal R-matrices, associated with the central extension (through the Drinfeld's double construction) of the quantum groups U_q(sl_n), are realized through an infinite dimensional spectral parameter dependent solution for the…

We study two classes of quantum spheres and hyperboloids which are $*$-quantum spaces for the quantum orthogonal group $\mathcal{O}(SO_q(3))$. We construct line bundles over the quantum homogeneous space of invariant elements for the…

The dimension of the moduli space of smooth pointed curves with prescribed Weierstrass semigroup at the market point is computed for three families of symmetric semigroups of multiplicity six. We also collect the dimensions of such moduli…

The "quantum-event / prime ideal in a category/ noncommutative-point" alternative to "classical-event / commutative prime ideal/ point" is suggested. Ideals in additive categories, prime spectra and representation of quivers are considered…

In this paper, we study the properties of Connes spectral distances between quantum states under unitary transformations. We mainly focus on spectral triples with matrix algebras acting on finite dimensional Hilbert spaces via some linear…

We study mirror symmetry (A-side vs B-side) in the framework of quantum differential systems. We focuse on the logarithmic and non-resonant case, which describes the geometric situation. We show that quantum differential systems provide a…

The structure of quantum principal bundles is studied, from the viewpoint of Tannaka-Krein duality theory. It is shown that if the structure quantum group is compact, principal G-bundles over a quantum space M are in a natural…

We discuss generalizations of the notion of i) the group of unitary elements of a (real or complex) finite dimensional C*-algebra, ii) gauge transformations and iii) (real) automorphisms, in the framework of compact quantum group theory and…

We study properties of a scalar quantum field theory on the two-dimensional noncommutative plane with $E_q(2)$ quantum symmetry. We start from the consideration of a firstly quantized quantum particle on the noncommutative plane. Then we…

We consider quantum invariants of 3-manifolds associated with arbitrary simple Lie algebras. Using the symmetry principle we show how to decompose the quantum invariant as the product of two invariants, one of them is the invariant…

Two hierarchies of quantum principal bundles over quantum real projective spaces are constructed. One hierarchy contains bundles with U(1) as a structure group, the other has the quantum group $SU_q(2)$ as a fibre. Both hierarchies are…

A class of quantum analogues of compact symmetric spaces of classical type is introduced by means of constant solutions to the reflection equations. Their zonal spherical functions are discussed in connection with $q$-orthogonal…

Building on the theory of noncommutative complex structures, the notion of a noncommutative K\"ahler structure is introduced. In the quantum homogeneous space case many of the fundamental results of classical K\"ahler geometry are shown to…

We define a quantum trace map from the skein module of a 3-manifold with torus boundary components to a module (left and right quotient of a quantum torus) constructed from an ideal triangulation. Our map is a 3-dimensional version of the…

Quantum homogeneous vector bundles are introduced by a direct description of their sections in the context of Woronowicz type compact quantum groups. The bundles carry natural topologies inherited from the quantum groups, and their sections…

The concept of a particle is ambiguous in quantum field theory. It is generally agreed that particles depend not only on spacetime, but also on coordinates used to parametrise spacetime points. One of us has in contrast proposed a…